Klaus Roth | |
|---|---|
 | |
| Born | Klaus Friedrich Roth (1925-10-29)29 October 1925 |
| Died | 10 November 2015(2015-11-10) (aged 90) Inverness, Scotland |
| Education | |
| Known for | |
| Awards |
|
| Scientific career | |
| Fields | Mathematics |
| Institutions | |
| Thesis | Proof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power (1950) |
| Doctoral advisor | Theodor Estermann |
| Other academic advisors | |
Klaus Friedrich RothFRS (29 October 1925 – 10 November 2015) was a German-born British mathematician who won theFields Medal for provingRoth's theorem on theDiophantine approximation ofalgebraic numbers. He was also a winner of theDe Morgan Medal and theSylvester Medal, and a Fellow of theRoyal Society.
Roth moved to England as a child in 1933 to escape the Nazis, and was educated at theUniversity of Cambridge andUniversity College London, finishing his doctorate in 1950. He taught at University College London until 1966, when he took a chair atImperial College London. He retired in 1988.
Beyond his work on Diophantine approximation, Roth made major contributions to the theory ofprogression-free sets inarithmetic combinatorics and to the theory ofirregularities of distribution. He was also known for his research onsums of powers, on thelarge sieve, on theHeilbronn triangle problem, and onsquare packing in a square. He was a coauthor of the bookSequences oninteger sequences.
Roth was born to a Jewish family inBreslau,Prussia, on 29 October 1925. His parents settled with him in London to escape Nazi persecution in 1933, and he was raised and educated in the UK.[1][2] His father, a solicitor, had been exposed to poison gas duringWorld War I and died while Roth was still young. Roth became a pupil atSt Paul's School, London from 1939 to 1943, and with the rest of the school he was evacuated from London toEasthampstead Park duringthe Blitz. At school, he was known for his ability in both chess and mathematics. He tried to join theAir Training Corps, but was blocked for some years for being German and then after that for lacking the coordination needed for a pilot.[2]
Roth read mathematics atPeterhouse, Cambridge, and playedfirst board for the Cambridge chess team,[2] finishing in 1945.[3]Despite his skill in mathematics, he achieved onlythird-class honours on theMathematical Tripos, because of his poor test-taking ability. His Cambridge tutor,John Charles Burkill, was not supportive of Roth continuing in mathematics, recommending instead that he take "some commercial job with a statistical bias".[2]Instead, he briefly became a schoolteacher atGordonstoun, between finishing at Cambridge and beginning his graduate studies.[1][2] While in Gordonstoun, he became a regular chess partner ofRobert Forbes Combe, who won the British championship the following year.[2]
On the recommendation ofHarold Davenport, he was accepted in 1946 to a master's program in mathematics atUniversity College London, where he worked under the supervision ofTheodor Estermann.[2] He completed a master's degree there in 1948, and a doctorate in 1950.[3] His dissertation wasProof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power.[4]
On receiving his master's degree in 1948, Roth became an assistant lecturer at University College London, and in 1950 he was promoted to lecturer.[5]His most significant contributions, on Diophantine approximation, progression-free sequences, and discrepancy, were all published in the mid-1950s,and by 1958 he was given the Fields Medal, mathematicians' highest honour.[2][6] However, it was not until 1961 that he was promoted to full professor.[1]During this period, he continued to work closely with Harold Davenport.[2]
He took sabbaticals at theMassachusetts Institute of Technology in the mid-1950s and mid-1960s, and seriously considered migrating to the United States.Walter Hayman andPatrick Linstead countered this possibility, which they saw as a threat to British mathematics, with an offer of a chair in pure mathematics atImperial College London, and Roth accepted the chair in 1966.[2] He retained this position until official retirement in 1988.[1] He remained at Imperial College as Visiting Professor until 1996.[3]
Roth's lectures were usually very clear but could occasionally be erratic.[2]TheMathematics Genealogy Project lists him as having only two doctoral students,[4] but one of them, William Chen, who continued Roth's work in discrepancy theory, became a Fellow of theAustralian Mathematical Society and head of the mathematics department atMacquarie University.[7]
In 1955, Roth married Mélèk Khaïry, who had attracted his attention when she was a student in his first lecture; Khaïry was a daughter of Egyptian senator Khaïry Pacha.[1][2] She came to work for the psychology department at University College London, where she published research on the effects of toxins on rats.[8]On Roth's retirement, they moved toInverness; Roth dedicated a room of their house to Latin dancing, a shared interest of theirs.[2][9]Khaïry died in 2002, and Roth died in Inverness on 10 November 2015 at the age of 90.[1][2][3]They had no children, and Roth dedicated the bulk of his estate, over one million pounds, to two health charities "to help elderly and infirm people living in the city of Inverness". He sent the Fields Medal with a smaller bequest to Peterhouse.[10]
Roth was known as a problem-solver in mathematics, rather than as a theory-builder. Harold Davenport writes that the "moral in Dr Roth's work" is that "the great unsolved problems of mathematics may still yield to direct attack, however difficult and forbidding they appear to be, and however much effort has already been spent on them".[6] His research interests spanned several topics innumber theory,discrepancy theory, and the theory ofinteger sequences.
The subject ofDiophantine approximation seeks accurate approximations ofirrational numbers byrational numbers. The question of how accuratelyalgebraic numbers could be approximated became known as the Thue–Siegel problem, after previous progress on this question byAxel Thue andCarl Ludwig Siegel. The accuracy of approximation can be measured by theapproximation exponent of a number, defined as the largest number such that has infinitely many rational approximations with. If the approximation exponent is large, then has more accurate approximations than a number whose exponent is smaller. The smallest possible approximation exponent is two: even the hardest-to-approximate numbers can be approximated with exponent two usingsimple continued fractions.[3][6] Before Roth's work, it was believed that the algebraic numbers could have a larger approximation exponent, related to thedegree of the polynomial defining the number.[2]
In1955, Roth published what is now known asRoth's theorem, completely settling this question. His theorem falsified the supposed connection between approximation exponent and degree, and proved that, in terms of the approximation exponent, the algebraic numbers are the least accurately approximated of any irrational numbers. More precisely, he proved that for irrational algebraic numbers, the approximation exponent is always exactly two.[3] In a survey of Roth's work presented byHarold Davenport to theInternational Congress of Mathematicians in 1958, when Roth was given the Fields Medal, Davenport called this result Roth's "greatest achievement".[6]
Another result called "Roth's theorem", from1953, is inarithmetic combinatorics and concernssequences of integers with no three in arithmetic progression. These sequences had been studied in 1936 byPaul Erdős andPál Turán, who conjectured that they must be sparse.[11][a]However, in 1942,Raphaël Salem andDonald C. Spencer constructed progression-free subsets of the numbers from to of size proportional to, for every.[12]
Roth vindicated Erdős and Turán by proving that it is not possible for the size of such a set to be proportional to: everydense set of integers contains a three-term arithmetic progression. His proof uses techniques fromanalytic number theory including theHardy–Littlewood circle method to estimate the number of progressions in a given sequence and show that, when the sequence is dense enough, this number is nonzero.[2][13]
Other authors later strengthened Roth's bound on the size of progression-free sets.[14] A strengthening in a different direction,Szemerédi's theorem, shows that dense sets of integers contain arbitrarily long arithmetic progressions.[15]
Although Roth's work on Diophantine approximation led to the highest recognition for him, it is his research on irregularities of distribution that (according to an obituary by William Chen andBob Vaughan) he was most proud of.[2] His1954 paper on this topic laid the foundations for moderndiscrepancy theory. It concerns the placement of points in a unit square so that, for every rectangle bounded between the origin and a point of the square, the area of the rectangle is well-approximated by the number of points in it.[2]
Roth measured this approximation by the squared difference between the number of points and times the area, and proved that for a randomly chosen rectangle theexpected value of the squared difference is logarithmic in. This result is best possible, and significantly improved a previous bound on the same problem byTatyana Pavlovna Ehrenfest.[16] Despite the prior work of Ehrenfest andJohannes van der Corput on the same problem, Roth was known for boasting that this result "started a subject".[2]
Some of Roth's earliest works included a1949 paper onsums of powers, showing thatalmost all positive integers could be represented as a sum of a square, a cube, and a fourth power, and a1951 paper on the gaps betweensquarefree numbers, describes as "quite sensational" and "of considerable importance" respectively by Chen and Vaughan.[2] His inaugural lecture at Imperial College concerned thelarge sieve: bounding the size of sets of integers from which manycongruence classes of numbers moduloprime numbers have been forbidden.[17] Roth had previously published a paper on this problem in1965.
Another of Roth's interests was theHeilbronn triangle problem, of placing points in a square to avoid triangles of small area. His1951 paper on the problem was the first to prove a nontrivial upper bound on the area that can be achieved. He eventually published four papers on this problem, the latest in1976.[18]Roth also made significant progress onsquare packing in a square. If unit squares are packed into an square in the obvious, axis-parallel way, then for values of that are just below an integer, nearly area can be left uncovered. AfterPaul Erdős andRonald Graham proved that a more clever tilted packing could leave a significantly smaller area, only,[19] Roth andBob Vaughan responded with a1978 paper proving the first nontrivial lower bound on the problem. As they showed, for some values of, the uncovered area must be at least proportionalto.[2][20]
In1966,Heini Halberstam and Roth published their bookSequences, oninteger sequences. Initially planned to be the first of a two-volume set, its topics included the densities of sums of sequences,bounds on the number of representations of integers as sums of members of sequences, density of sequences whose sums represent all integers,sieve theory and theprobabilistic method, andsequences in which no element is a multiple of another.[21] A second edition was published in 1983.[22]
Roth won theFields Medal in 1958 for his work on Diophantine approximation. He was the first British Fields medallist.[1] He was elected to theRoyal Society in 1960, and later became an Honorary Fellow of theRoyal Society of Edinburgh, Fellow of University College London, Fellow of Imperial College London, and Honorary Fellow of Peterhouse.[1] It was a source of amusement to him that his Fields Medal, election to the Royal Society, and professorial chair came to him in the reverse order of their prestige.[2]
TheLondon Mathematical Society gave Roth theDe Morgan Medal in 1983.[3]In 1991, the Royal Society gave him theirSylvester Medal "for his many contributions to number theory and in particular his solution of the famous problem concerning approximating algebraic numbers by rationals."[23]
Afestschrift of 32 essays on topics related to Roth's research was published in 2009, in honour of Roth's 80th birthday,[24]and in 2017 the editors of theUniversity College London journalMathematika dedicated a special issue to Roth.[25]After Roth's death, the Imperial College Department of Mathematics instituted the Roth Scholarship in his honour.[26]
{{cite journal}}: CS1 maint: untitled periodical (link){{cite journal}}: CS1 maint: untitled periodical (link){{cite journal}}: CS1 maint: untitled periodical (link){{cite journal}}: CS1 maint: untitled periodical (link)
